Max Planck · Verhandlungen der Deutschen Physikalischen Gesellschaft 2 (1900): 237–245 · presented 14 December 1900
Gentlemen: some weeks ago I had the honour to draw your attention to a new formula which seemed to me suited to express the law of the distribution of radiation energy over the whole range of the normal spectrum. I remarked then that, in my opinion, the usefulness of this equation was not based solely on its close agreement with the observations available, but rested chiefly on the simple structure of the formula. Today I should like to lay before you the theoretical deduction of that formula.
To obtain it I follow the lead of Boltzmann: the entropy S of a system in a given state is proportional to the logarithm of the probability W of that state. Of the N resonators of frequency ν in the cavity, let the total energy be E. The whole question reduces to determining the probability W — the number of ways in which this energy can be distributed over the N resonators.
The energy element
We consider, however — this is the most essential point of the whole calculation — E to be composed of a very definite number of equal parts, and use thereto the constant of nature h = 6.55 × 10⁻²⁷ erg·sec. This constant, multiplied by the common frequency ν of the resonators, gives us the energy element ε in ergs, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.
The probability W is then the number of ways in which the P indistinguishable energy elements can be distributed over the N resonators. The energy of a resonator is thus to be regarded as made up of a whole number of finite equal parts, and not as a continuously divisible quantity.
ε = hν.
I therefore call h the elementary quantum of action. It is the presence of this finite element — energy exchanged not continuously, but in whole multiples of hν — that makes the calculation succeed where the classical, continuous treatment had failed.
The radiation law
Carrying the calculation through, the energy distribution of the normal spectrum follows, per unit frequency, as u = (8πhν³/c³) / (e^{hν/kT} − 1). With its help it is possible to derive a radiation formula in complete agreement with the observations made up to the present.
Comparison with the measurements of Kurlbaum and of Lummer and Pringsheim fixes the two constants as h = 6.55 × 10⁻²⁷ erg·sec and k = 1.346 × 10⁻¹⁶ erg·degree⁻¹. From k there follow, in turn, the number of molecules in a gramme-molecule and the elementary electric charge, e = 4.69 × 10⁻¹⁰ electrostatic units — values in satisfactory agreement with those obtained by other means.
Berlin · 14 December 1900